Eigenvalue and Stability Problems in Applied Mathematics

应用数学中的特征值和稳定性问题

• 批准号: 0807584
• 负责人: Jared Bronski
• 金额: $ 14.6万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0807584&HistoricalAwards=false
• 关键词: Eigenvalue; Stability; Problems; Applied; Mathematics

The research that will be supported by this award addresses a number of topics in applied mathematics that have the common theme of eigenvalue problems. Eigenvalue problems arise in a number of contexts in various branches of science, in particular in questions related to the stability of a physical system. Specifically, in many equations arising in mathematical physics there may exist valid exact solutions which are unstable, in the sense that nearby solutions rapidly diverge from this exact solutions as time progresses. Such solutions are often difficult or impossible to realize in an experiment, since the initial conditions must be chosen very precisely in order for the solutions to remain observable for any appreciable length of time. Put differently, physically interesting solutions typically are the stable ones. and what is not. Unfortunately it is often very difficult to determine whether a particular solution is stable or not. The problems that will be studied under this award range from water waves and wave propagation in photonic materials to models in physiology. Specifically, problems related to photonic crystals with defects, standing waves for nonlinear Schroedinger equations, vortex crystals for Bose-Einstein condensates, scattering problems related to the Sine-Gordon equations, and a problem from neurophysiology (oculomotor integrator of the brain) will studied. In some of these projects, geometric methods for establishing the instability of solutions will be developed. These methods are very general, being based on geometric considerations rather than details of the equation in question. Thus they are potentially applicable to problems in many disciplines within science. Other techniques such as asymptotics and topological arguments will also be employed. Stability is a fundamental property of many physical systems. For example, a pocket watch hanging from a chain is a stable system - it will eventually return to its rest state when perturbed. A pencil that is balancing on its tip on a table is unstable - a very small perturbation will cause it to fall over. For many engineered physical systems, it is however not clear if it is stable or not. Examples occur in the design of novel optical materials, where the stability of a traveling light pulse is of interest. In other situations, e.g. in physiology, instability is related to malfunction, and it is of interest to explore mechanisms that lead to instability. For example, the oculomotor integrator, which is part of the brain subsystem that moves the eyes, is known to function properly as long as a mathematical model satisfies a suitable condition (a single dominant eigenvalue is near the origin). The system becomes unstable if there are eigenvalues in locations where they not are supposed to be, resulting in a condition known as congenital nystagmus that effects one in about 2000 people. The research in this project is primarily concerned with questions of stability in the broadest sense. It will provide tools that are applicable to problems ranging from photonics to physiology.

该奖项将支持的研究涉及应用数学中的许多主题，这些主题具有特征值问题的共同主题。特征值问题出现在许多科学分支的背景下，特别是与物理系统的稳定性有关的问题。具体地说，在数学物理中出现的许多方程中，可能存在有效的不稳定精确解，在这个意义上，随着时间的推移，附近的解迅速偏离这个精确解。这样的解通常很难或不可能在实验中实现，因为初始条件必须非常精确地选择，以便解在任何可观的时间长度内保持可观察。换句话说，物理上有趣的解决方案通常是稳定的。 什么不是不幸的是，通常很难确定一个特定的解决方案是否稳定。该奖项将研究的问题范围从水波和光子材料中的波传播到生理学模型。具体来说，与具有缺陷的光子晶体相关的问题、非线性薛定谔方程的驻波、玻色-爱因斯坦凝聚体的涡旋晶体、与Sine-Gordon方程相关的散射问题以及神经生理学问题（大脑的眼球运动积分器）将研究。在其中一些项目中，将开发确定解的不稳定性的几何方法。这些方法是非常普遍的，是基于几何的考虑，而不是细节的方程的问题。因此，它们可能适用于科学中许多学科的问题。其他技术，如渐近和拓扑参数也将采用。稳定性是许多物理系统的基本属性。例如，挂在链子上的怀表是一个稳定的系统-当受到扰动时，它最终会返回到静止状态。一只铅笔在桌子上保持笔尖的平衡是不稳定的--一个很小的扰动就会导致它翻倒。然而，对于许多工程物理系统来说，它是否稳定还不清楚。例如，在设计新型光学材料时，人们对行进光脉冲的稳定性很感兴趣。在其他情况下，例如在生理学中，不稳定性与功能障碍有关，并且探索导致不稳定性的机制是有意义的。例如，眼球运动积分器是大脑子系统的一部分，它移动眼睛，只要数学模型满足适当的条件（单个主特征值靠近原点），它就可以正常工作。如果特征值出现在不应该出现的位置，系统就会变得不稳定，导致一种被称为先天性眼球震颤的情况，大约每2000人中就有一人受到影响。本项目的研究主要涉及最广义的稳定性问题。它将提供适用于从光子学到生理学的问题的工具。